Advanced TDDFT II Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York Memory-Dependence in Linear Response a. Double.

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Presentation on theme: "Advanced TDDFT II Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York Memory-Dependence in Linear Response a. Double."— Presentation transcript:

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Advanced TDDFT II Neepa T. Maitra Hunter College and the Graduate Center of the City University of New York Memory-Dependence in Linear Response a. Double Excitations b. Charge Transfer Excitations f xc

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Double (Or Multiple) Excitations  – poles at true states that are mixtures of singles, doubles, and higher excitations  S -- poles at single KS excitations only, since one-body operator can’t connect Slater determinants differing by more than one orbital.    has more poles than  s ? How does f xc generate more poles to get states of multiple excitation character? Consider: How do these different types of excitations appear in the TDDFT response functions?

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An Exercise! Deduce something about the frequency-dependence required for capturing states of triple excitation character – say, one triple excitation coupled to a single excitation.

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Diagonalize many-body H in KS subspace near the double-ex of interest, and require reduction to adiabatic TDDFT in the limit of weak coupling of the single to the double: N.T. Maitra, F. Zhang, R. Cave, & K. Burke JCP 120, 5932 (2004) usual adiabatic matrix element dynamical (non-adiabatic) correction Practical Approximation for the Dressed Kernel So: (i) scan KS orbital energies to see if a double lies near a single, (ii)apply this kernel just to that pair (iii)apply usual ATDDFT to all other excitations

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ATDDFT fundamentally fails to describe double-excitations: strong frequency-dependence is essential. Diagonalizing in the (small) subspace where double excitations mix with singles, we can derive a practical frequency-dependent kernel that does the job. Shown to work well for simple model systems, as well as real molecules. Likewise, in autoionization, resonances due to double-excitations are missed in ATDDFT. Summary on Doubles Next: Long-Range Charge-Transfer Excitations

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A Useful Exercise! To deduce the step in the potential in the bonding region between two open-shell fragments at large separation: Take a model molecule consisting of two different “one-electron atoms” (1 and 2) at large separation. The KS ground-state is the doubly-occupied bonding orbital: where    r) and n(r ) =  1 2 (r) +  2 2 (r) is the sum of the atomic densities. The KS eigenvalue  0 must =   =  I  where I 1 is the smaller ionization potential of the two atoms. Consider now the KS equation for r near atom 1, where and again for r near atom 2, where Noting that the KS equation must reduce to the respective atomic KS equations in these regions, show that v s, must have a step of size       = I 2 –I 1 between the atoms.

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So far for our model: Discussed step and peak structures in the ground-state potential of a dissociating molecule : hard to model, spatially non-local Fundamentally, these arise due to the single-Slater-determinant description of KS (one doubly-occupied orbital) – the true wavefunction, requires minimally 2 determinants (Heitler-London form) In practise, could treat ground-state by spin-symmetry breaking  good ground-state energies but wrong spin-densities See Dreissigacker & Lein, Chem. Phys. (2011) - clever way to get good DFT potentials from inverting spin-dft Next: What are the consequences of the peak and step beyond the ground state? Response and Excitations

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What about TDDFT excitations of the dissociating molecule? Recall the KS excitations are the starting point; these then get corrected via f xc to the true ones. LUMO HOMO  ~ e -cR Near-degenerate in KS energy “Li”“H” Step  KS molecular HOMO and LUMO delocalized and near-degenerate But the true excitations are not! Find: The step induces dramatic structure in the exact TDDFT kernel ! Implications for long-range charge-transfer. Static correlation induced by the step!

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Important difference between (closed-shell) molecules composed of (i)open-shell fragments, and (ii)those composed of closed-shell fragments. HOMO delocalized over both fragments HOMO localized on one or other  Revisit the previous analysis of CT problem for open-shell fragments: Eg. apply SMA (or SPA) to HOMO  LUMO transition But this is now zero ! q = bonding  antibonding Now no longer zero – substantial overlap on both atoms. But still wrong. Wait!! !! We just saw that for dissociating LiH-type molecules, the HOMO and LUMO are delocalized over both Li and H  f xc contribution will not be zero!